Theorem opnnei 21138

Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)

Assertion

Ref	Expression
opnnei	⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei

Step	Hyp	Ref	Expression
1		0opn 20922	. . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
2	1	adantr 468	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3		eleq1 2873	. . . . 5 ⊢ (𝑆 = ∅ → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
4	3	adantl 469	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
5	2, 4	mpbird 248	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆 ∈ 𝐽)
6		rzal 4268	. . . 4 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
7	6	adantl 469	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
8	5, 7	2thd 256	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9		opnneip 21137	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
10	9	3expia 1143	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑥 ∈ 𝑆 → 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
11	10	ralrimiv 3153	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
12	11	ex 399	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
13	12	adantr 468	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14		df-ne 2979	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15		r19.2z 4255	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
16	15	ex 399	. . . . . 6 ⊢ (𝑆 ≠ ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
17	14, 16	sylbir 226	. . . . 5 ⊢ (¬ 𝑆 = ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18		eqid 2806	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	neii1 21124	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ⊆ ∪ 𝐽)
20	19	ex 399	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
21	20	rexlimdvw 3222	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
22	17, 21	sylan9r 500	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
23	18	ntrss2 21075	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
24	23	adantr 468	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25		vex 3394	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
26	25	snss 4506	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
27	26	ralbii 3168	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28		dfss3 3787	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
29	28	biimpri 219	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
30	29	adantl 469	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
31	27, 30	sylan2br 584	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
32	24, 31	eqssd 3815	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
33	32	ex 399	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
34	25	snss 4506	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 ↔ {𝑥} ⊆ 𝑆)
35		sstr2 3805	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊆ 𝑆 → (𝑆 ⊆ ∪ 𝐽 → {𝑥} ⊆ ∪ 𝐽))
36	35	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ∪ 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
37	36	adantl 469	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
38	34, 37	syl5bi 233	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
39	38	imp 395	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → {𝑥} ⊆ ∪ 𝐽)
40	18	neiint 21122	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41	40	3com23 1149	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42	41	3expa 1140	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
43	39, 42	syldan 581	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
44	43	ralbidva 3173	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
45	18	isopn3 21084	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
46	33, 44, 45	3imtr4d 285	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
47	46	ex 399	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽)))
48	47	com23 86	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
49	48	adantr 468	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
50	22, 49	mpdd 43	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
51	13, 50	impbid 203	. 2 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
52	8, 51	pm2.61dan 838	1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637   ∈ wcel 2156   ≠ wne 2978  ∀wral 3096  ∃wrex 3097   ⊆ wss 3769  ∅c0 4116  {csn 4370  ∪ cuni 4630  ‘cfv 6101  Topctop 20911  intcnt 21035  neicnei 21115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-top 20912  df-ntr 21038  df-nei 21116

This theorem is referenced by:  neiptopreu  21151  flimcf  21999